Let σ = (24)(1759)(428)(735) ∈ S9.
1. Write σ as a product of disjoint cycles

To write the permutation \(\sigma = (24)(1759)(428)(735)\) as a product of disjoint cycles, we will track the movement of each element in the permutation.

Let's start with the smallest element, \(1\):
- \(1 \rightarrow 7\) (from the cycle \((1759)\))
- \(7 \rightarrow 3\) (from the cycle \((735)\))
- \(3 \rightarrow 3\) (stays the same because it's not affected by any other cycles)

So, \(1 \rightarrow 3\).

Now let's track the element \(2\):
- \(2 \rightarrow 4\) (from the cycle \((24)\))
- \(4 \rightarrow 2\) (from the cycle \((428)\))

So, \(2 \rightarrow 2\), meaning \(2\) is fixed.

Now for \(3\):
- We already saw that \(3\) goes to \(3\).

Now for \(4\):
- \(4 \rightarrow 2\) (from the cycle \((428)\))
- \(2 \rightarrow 4\) (from the cycle \((24)\))

So, \(4 \rightarrow 4\), meaning \(4\) is fixed.

Next, let's track \(5\):
- \(5 \rightarrow 9\) (from the cycle \((1759)\))
- \(9 \rightarrow 9\) (fixed as it's not affected by any other cycles)

So, \(5 \rightarrow 9\).

Now for \(7\):
- \(7 \rightarrow 3\) (from the cycle \((735)\))

We already know the result from \(1 \rightarrow 7\) and \(7 \rightarrow 3\).

Finally, let's track \(8\):
- \(8 \rightarrow 8\) (fixed because it does not change in the cycle \((428)\))

So, \(8\) is fixed.

Putting this all together, the disjoint cycles representation of \(\sigma\) is:
\[
\sigma = (1 \, 7 \, 3 \, 5 \, 9)
\]

This is because:
- \(1\) maps to \(7\), \(7\) maps to \(3\), \(3\) maps to \(5\), \(5\) maps to \(9\), and \(9\) maps back to \(1\).
- \(2\), \(4\), \(6\), and \(8\) are fixed and are not part of any nontrivial cycles.

**Summary:** The permutation \(\sigma\) in disjoint cycle notation is:
\[
\sigma = (1 \, 7 \, 3 \, 5 \, 9)
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How do you identify fixed points in a permutation?
2. What is the order of a permutation?
3. Can every permutation be written as a product of disjoint cycles?
4. How do you find the inverse of a permutation?
5. How can you determine if a permutation is even or odd?

**Tip:** When writing permutations as disjoint cycles, remember that disjoint cycles commute, meaning their order in the product does not affect the result.

Learn how to write the permutation σ = (24)(1759)(428)(735) as a product of disjoint cycles with a detailed step-by-step solution. Understand the movement of elements in the permutation and identify fixed points. Suitable for advanced high school students studying permutations and cycles in abstract algebra.

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